Trinomial Calculator
Factor any trinomial, expand binomials with FOIL, evaluate at a given x, or find the discriminant and roots. Step-by-step solutions for every operation.
Operation
Quick Examples
Enter Coefficients
Enter values for ax² + bx + c
Trinomial: x² + 5x + 6
Step-by-Step Solution
Factored Form
Discriminant: 1
Roots: x = -2, -3
Reference
Common Factorable Trinomials
| Trinomial | Factored Form | Roots | Type |
|---|---|---|---|
| x² + 5x + 6 | (x + 2)(x + 3) | x = -2, -3 | Simple (a=1) |
| x² - 7x + 12 | (x - 3)(x - 4) | x = 3, 4 | Simple (a=1) |
| x² - x - 6 | (x - 3)(x + 2) | x = 3, -2 | Simple (a=1) |
| 2x² + 7x + 3 | (2x + 1)(x + 3) | x = -1/2, -3 | Leading coeff |
| 6x² + 11x - 10 | (2x + 5)(3x - 2) | x = -5/2, 2/3 | Leading coeff |
| x² + 6x + 9 | (x + 3)² | x = -3 (double) | Perfect square |
| 4x² - 12x + 9 | (2x - 3)² | x = 3/2 (double) | Perfect square |
| x² - 9 | (x + 3)(x - 3) | x = 3, -3 | Difference of squares |
| x² + 2x + 5 | Prime (no real factors) | Complex | Prime |
How to Factor a Trinomial
Trinomial Factoring and Operations Guide
A trinomial is any expression with exactly three terms. In algebra, the term almost always refers to a quadratic trinomial: something in the form ax² + bx + c. Factoring these expressions is one of the core skills in algebra because it unlocks the ability to solve quadratic equations, simplify rational expressions, and graph parabolas.
Simple Trinomials (a = 1)
When the leading coefficient is 1, factoring is a matter of finding two numbers that multiply to c and add to b. For x² + 5x + 6, you need two numbers that multiply to 6 and add to 5. That would be 2 and 3, giving (x + 2)(x + 3). This works because when you FOIL the binomials back out, the outer and inner products combine to recreate the middle term.
The AC Method
When a is not 1, things get trickier. The AC method handles this. Multiply a times c to get the "AC product." Then find two numbers that multiply to that product and add to b. Use those numbers to split the middle term into two pieces, creating a four-term expression you can factor by grouping.
Take 2x² + 7x + 3. The AC product is 2 times 3 = 6. You need two numbers that multiply to 6 and add to 7. That is 1 and 6. Rewrite as 2x² + x + 6x + 3, then group: x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).
Perfect Square Trinomials
A perfect square trinomial factors into a single binomial squared. The pattern is a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)². To spot one, check if the first and last terms are perfect squares, and if the middle term is exactly twice the product of their square roots. x² + 6x + 9 fits because 9 = 3² and 6 = 2(1)(3).
The Discriminant
The discriminant b² - 4ac tells you everything about factorability before you start. If it is a perfect square (like 0, 1, 4, 9, 16...), the trinomial factors over the integers. If it is positive but not a perfect square, the roots are irrational and the trinomial is prime over the integers. If it is negative, there are no real roots at all.
FOIL and Expanding
FOIL is factoring in reverse. When you multiply (ax + b)(cx + d), the four products are: First (ac)x², Outer (ad)x, Inner (bc)x, Last (bd). Combine the two middle terms to get a trinomial. This calculator handles the expansion automatically and shows each step so you can verify your manual work.
When Factoring Fails
Not every trinomial factors neatly. When the discriminant is not a perfect square, the quadratic formula still gives you the roots, but they involve square roots and are not rational numbers. In those cases, the trinomial is called "prime over the rationals." You can still solve the equation; you just cannot write it as a product of integer-coefficient binomials.